Waiter-Client Clique-Factor Game

TL;DR

This paper analyzes the duration of a combinatorial game involving Waiter and Client on a complete graph, establishing bounds on the number of rounds until Waiter forces a $K_k$-factor in Client's graph with both players playing optimally.

Contribution

It provides the first non-trivial lower bound on the game duration for large $k$, complementing existing upper bounds and advancing understanding of strategic graph game dynamics.

Findings

Lower bound: $2^{k/3-o(k)}n$ rounds

Upper bound: $2^k rac{n}{k} + C(k)$ rounds

Both bounds are tight up to exponential factors

Abstract

Fix two integers $n, k$, with $n$ divisible by $k$, and consider the following game played by two players, Waiter and Client, on the edges of $K_n$. Starting with all the edges marked as unclaimed, in each round, Waiter picks two yet unclaimed edges. Client then chooses one of these edges to be added to Client's graph, while the other edge is added to Waiter's graph. Waiter wins if she eventually forces Client to create a $K_k$-factor in Client's graph. If she does not manage to do that, Client wins. For fixed $k$ and large enough $n$, it can be easily shown that Waiter wins if she plays optimally (in particular, this is an immediate consequence of our result that for such $n$, Waiter can win quite fast). The question posed by Clemens et al. is how long the game will last if Waiter aims to win as fast as she can, Client tries to delay her as much as he can, and they both play optimally. We denote this optimal number of rounds by $\tau_{WC}(\mathcal{F}_{n,K_k-\text{fac}},1 ) $. In the present paper, we obtain the first non-trivial lower bound on this quantity for large $k$. Together with a simple upper bound following the strategy of Clemens et al., this gives $2^{k/3-o(k)}n \leq \tau_{WC}(\mathcal{F}_{n,K_k-\text{fac}},1 ) \leq 2^k\frac{n}{k}+C(k)$, where $C(k)$ is a constant dependent only on $k$ and the $o(k)$ term is independent of $n$ as well.